Abstract
Lakoff and Núñez have argued that all of mathematics is a conceptual system created through metaphors on the basis of the ideas and modes of reasoning grounded in the sensory motor system. This paper explores this view by means of a Lakatosian reconstruction of the history and prehistory of the intermediate-value theorem, in which the notion of continuity plays an essential role. I conclude that in order to give an acceptable description of the actual development of mathematics, Lakoff’s and Núñez’s view must be amended: Mathematics can be viewed as a system of conceptual metaphors; however, it is permanently refined through proofs and refutations.
Parts of this paper are based on Koetsier (1995).
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Notes
- 1.
Quoted in Lakatos (1961/1973, p. 73).
- 2.
Spalt (1988) also deals with the history of the intermediate value theorem.
- 3.
Cf. Müller and Kronert (1969, p. 136).
- 4.
Leibniz (1695, p. 284).
- 5.
Leibniz (1695, p. 284).
- 6.
Leibniz wrote Y, Z and Y+Z, in this sentence, but must have meant Y, Z and Y+Z.
- 7.
Leibniz (1695, pp. 284-285)
- 8.
Lagrange (1808, pp. 1–2)
- 9.
Lagrange (1808, pp. 101–102)
- 10.
In the following reconstruction I will interpret some of Cauchy’s results in accordance with the traditional view of his work. A good presentation of this view is in Grabiner (1981). Grabiner nicely shows that the idea that the notion of limit is related to methods of approximation played an important heuristic role in Cauchy’s foundational work. For a rather different view of Cauchy’s foundational work in analysis see Spalt (1996).
- 11.
I am not the first one to suggest that Cauchy’s definition of continuity was born here. Cf. Daval and Guilbaud (1945, p. 117).
- 12.
Cauchy (1821, pp. 115–116)
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Acknowledgements
I am grateful to Brendan Larvor and to two anonymous referees for commenting on an earlier version of this paper.
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Koetsier, T. (2010). Lakatos, Lakoff and Núñez: Towards a Satisfactory Definition of Continuity. In: Hanna, G., Jahnke, H., Pulte, H. (eds) Explanation and Proof in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0576-5_3
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